Correlated disorder as a way towards robust superconductivity

The effect of disorder on superconductivity has been a subject of extensive research. Early theoretical work by Abrikosov and Gor’kov, and Anderson, established that *s*-wave superconductivity is relatively insensitive to weak, uncorrelated nonmagnetic disorder. However, experimental findings consistently demonstrate that strong disorder suppresses superconductivity, potentially leading to a superconductor-insulator transition. A fascinating intermediary regime exists where disorder can enhance superconductivity, but most theoretical studies have focused on uncorrelated, random disorder. This study departs from this assumption, recognizing that in real materials, impurities and defects exhibit spatial correlations. This work hypothesizes that these correlations profoundly affect the superconducting state, making it more robust and less sensitive to disorder. This is of paramount importance for developing novel superconducting materials and functionalities. The researchers aim to explore the interplay between order and disorder to optimize superconducting properties, focusing on how long-range correlations of the disorder potential influence superconductivity at zero temperature. This investigation uses a 2D model with spatial correlations in the disorder potential and examines the superconducting properties, including the order parameter, its statistical distribution, and superfluid stiffness.

Numerous studies have explored the influence of disorder on superconductivity. While weak disorder has a negligible effect on *s*-wave superconductors, strong disorder leads to suppression and even a superconductor-insulator transition. This transition is linked to phenomena such as Anderson localization, phase fluctuations, and modifications to the electronic density of states. The discovery of quasi-2D materials has amplified these effects due to their reduced dimensionality. Interestingly, certain studies have reported disorder-induced enhancement of superconductivity, linked to increased inhomogeneity and local pairing correlations. These enhancements have been observed in diverse systems, including quasi-1D MoSe chains, TaSx monolayers, and NbSe2 monolayers. Theoretical explanations often involve disorder-induced multifractal structures in electronic wavefunctions. Previous theoretical models, however, largely simplified the disorder as spatially uncorrelated. This contrasts with real materials exhibiting inherent structural correlations in defect distributions. This paper builds on this gap in knowledge, focusing on correlated disorder to gain a more realistic understanding of the complex relationship between disorder and superconductivity.

The researchers employed a two-dimensional (2D) model of a disordered superconductor to analyze the effects of correlated disorder. The spatial correlations of the disorder potential in reciprocal space followed a power law, S_V(q) ∝ q^-α, where the exponent α controls the correlation degree. α = 0 represents completely uncorrelated disorder, while increasing α introduces long-range correlations. The correlation length ξ_v was calculated. The analysis involved solving the Bogoliubov-de Gennes (BdG) equations for the superconducting order parameter Δ using numerically generated disorder potentials with various α values. The spatial profiles of the disorder potential and order parameter were visualized. The correlation between the disorder potential and order parameter was quantified using a statistical correlator. Statistical properties of the order parameter were analyzed by examining its distribution, fitting it to a log-normal function to quantify deviations from this behavior in the presence of correlated disorder. Spatial correlations of the order parameter were investigated through a correlation function and its Fourier transform to identify characteristic length scales. Finally, the superfluid stiffness D was calculated to assess the global superconductivity, including both mean-field and fluctuation-corrected values, considering the diamagnetic and paramagnetic contributions. The calculations were performed on a 40x40 lattice with periodic boundary conditions, averaged over 50 independent disorder realizations to obtain statistically reliable results. The model used the BCS Hamiltonian with an on-site interaction term and incorporated a disordered potential. The Hartree self-consistency equation was included to account for the effect of inhomogeneities in strong disorder. The numerical parameters were chosen to represent a weak coupling BCS superconductor.

The study reveals a significant impact of long-range correlations on superconducting properties. Visual inspection of spatial profiles of both the disorder potential and the order parameter showed that increased correlation (higher α) leads to a more homogeneous order parameter, suppressing the formation of normal state (N) domains within the superconducting (S) phase. This indicates a notable enhancement of superconductivity. The correlation between the disorder potential and the order parameter is quantified and found to increase with α, meaning that superconductivity is more likely in areas with low disorder potential when the disorder is more correlated. Analysis of the order parameter's statistical distribution shows that while it follows a log-normal distribution for uncorrelated disorder, deviations arise for correlated disorder, particularly at high disorder strength. Specifically, the distribution shifts towards larger values of the order parameter with increasing α, suggesting enhanced superconductivity. The spatial correlations of the order parameter are also enhanced by increased α. The superfluid stiffness, a measure of global superconductivity, was calculated, accounting for both mean-field and phase fluctuation effects. The results consistently demonstrate that higher α values lead to a higher superfluid stiffness and thus more robust superconductivity, even at higher disorder strengths. The critical disorder strength at which the superfluid stiffness becomes zero (superconductivity vanishes) increases with α. The analysis of the diamagnetic and paramagnetic contributions to the stiffness shows that the paramagnetic component is particularly affected by the correlation, exhibiting increased values at higher α, further confirming the enhancement of superconductivity. A phase diagram mapping the superfluid stiffness (both mean-field and fluctuation-corrected) against disorder strength and correlation degree highlights the robust enhancement of superconductivity by correlated disorder.

The findings directly address the research question by demonstrating that spatially correlated disorder substantially affects superconductivity. Contrary to the traditional understanding based on uncorrelated disorder models, this study shows that correlated disorder can significantly enhance superconductivity, highlighting the importance of considering the spatial arrangement of impurities in material design. The increased homogeneity of the order parameter with correlated disorder indicates a more robust superconducting state, which is further confirmed by the enhanced superfluid stiffness. This challenges the conventional view that strong disorder always suppresses superconductivity and establishes correlations as a key parameter in controlling superconducting properties. The results are consistent with the intuitive expectation that superconductivity is favored in regions of low disorder potential. The interplay between the length scales of the BCS coherence length, the disorder correlation length, and the size of superconducting clusters is crucial in determining the net effect of correlated disorder. The observed deviations from the log-normal distribution of the order parameter for correlated disorder underscore the limitations of models based solely on uncorrelated disorder. This work provides a deeper understanding of the complex interplay between disorder and superconductivity, paving the way for more sophisticated materials design strategies.

This study presents compelling evidence for the substantial role of spatially correlated disorder in influencing superconductivity. The enhancement of superconducting properties observed with increasing correlation highlights the importance of considering not only the amount of disorder but also its spatial arrangement. The methodology employed, combining numerical solutions of the BdG equations with statistical analysis of the order parameter and superfluid stiffness, provides a robust framework for investigating this complex phenomenon. Future research could explore the impact of different correlation functions beyond the power-law model employed here, investigate the influence of temperature, and extend the analysis to other pairing symmetries. Understanding the relationship between disorder correlations and superconductivity opens new avenues for designing materials with enhanced superconducting properties.

The study focuses solely on zero-temperature behavior, limiting the generalizability to finite-temperature scenarios where thermal fluctuations would play a role. The model utilizes a specific form of correlated disorder, a power-law correlation function in reciprocal space, so the results might not be directly transferable to all types of correlated disorder found in real materials. The computational cost limited the size of the simulated system, so finite-size effects could have influenced the results, although measures were taken to mitigate this. Finally, the model neglected long-range Coulomb interactions, which could play a role in some disordered superconductors.